Strong coupling of finite element methods for the Stokes-Darcy problem

TL;DR

This paper develops a stable finite element method for the coupled Stokes-Darcy problem, ensuring mass conservation and optimal convergence by combining compatible finite element spaces for both flow regimes.

Contribution

It introduces a systematic approach to combine mixed finite elements for Darcy and Stokes flows with exact transmission conditions, ensuring stability and convergence.

Findings

The proposed method achieves quasi-optimal convergence rates.

Numerical tests confirm theoretical stability and accuracy.

The approach handles the transmission condition exactly in the velocity space.

Abstract

The aim of this paper is to propose a systematic way to obtain convergent finite element schemes for the Darcy-Stokes flow problem by combining well-known mixed finite elements that are separately convergent for Darcy and Stokes problems. In the approach in which the Darcy problem is set in its natural $\mathbf{H}(\text{div})$ formulation and the Stokes problem is expressed in velocity-pressure form, the transmission condition ensuring global mass conservation becomes essential. As opposed to the strategy that handles weakly this transmission condition through a Lagrange multiplier, we impose here this restriction exactly in the space of global velocity field. Our analysis of the Galerkin discretization of the resulting problem reveals that, if the mixed finite element space used in the Darcy domain admits an $\mathbf{H}(\text{div})$-stable discrete lifting of the normal trace, then it can be combined with any stable Stokes mixed finite element of the same order to deliver a stable global method with quasi-optimal convergence rate. Finally, we present a series of numerical tests confirming our theoretical convergence estimates.